Welcome
I can not help with much first hand knowledge or south Texas but here is one suggestion and some parks that have hikes
there is a book called hiking and backtracking trails of Texas by Mickey Little its at least in its fifth edition now.that is my number one resourch for our family and the scouts
Your describing backpacking if you want to hike in, eat, sleep and hike out, that is not the easiest thing to do if you have not done it.
Gear (stoves, shoes and packs are the most important here) and weight distribution of food,gear and water is very important especially with children.
The backpaking park closest to your circle are Choke Canyon SP (says 18 miles of trails)there are 2 units.
Hill Country SNA and Bastrop and Lost Maples all have backpacking trails
If you have more questions let us chip in
see this link
TPWD: Campsites